2.15.6 Boussinesq type $u_{tt}-u_{xx}-2\alpha (u{u}_x{)}_x-\beta u_{xxtt}=0$

problem number 116

Added December 27, 2018.

Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations

Boussinesq type PDE. Solve for $u(x,t)$ $$u_{tt}-u_{xx}-2\alpha (u{u}_x{)}_x-\beta u_{xxtt}=0$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] - D[u[x, t], {x, 4}] - 3*D[u[x, t]^2, {x, 2}] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

$$\left\{\{u(x,t)\to \frac{1}{6}(-12c_1{}^2{\tanh }^2(c_{2}t+c_{1}x+c_3)+8c_1{}^2+\frac{c_2{}^2}{c_1{}^2}-1)\}\right\}$$

Maple ✓

restart; 
pde := diff(u(x,t),t$2)-diff(u(x,t),x$2)-2*alpha*diff( (u(x,t)*diff(u(x,t),x)) ,x) - beta*diff(u(x,t),x,x,t,t)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
 

$$u(x,t)=1/2\frac{-12{\mathit{\_}\mathit{C}\mathit{3}}^2\beta {(\tanh (\mathit{\_}\mathit{C}\mathit{2}x+\mathit{\_}\mathit{C}\mathit{3}t+\mathit{\_}\mathit{C}\mathit{1}))}^2{\mathit{\_}\mathit{C}\mathit{2}}^2+(8{\mathit{\_}\mathit{C}\mathit{3}}^2\beta -1){\mathit{\_}\mathit{C}\mathit{2}}^2+{\mathit{\_}\mathit{C}\mathit{3}}^2}{\alpha {\mathit{\_}\mathit{C}\mathit{2}}^2}$$

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